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Numerical simulation of the generalized Huxley equation by He’s variational iteration method
Applied Mathematics and Computation 186 (2007) 1322–1325 www.elsevier.com/locate/amc
Numerical simulation of the generalized Huxley equation by He’s variational iteration method B. Batiha *, M.S.M. Noorani, I. Hashim School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, Malaysia
Abstract By means of variational iteration method the solution of generalized Huxley equation are obtained, comparison with the Adomian decomposition method is made, showing that the former is more effective than the later. In this paper, He’s variational iteration method is introduced to overcome the difficulty arising in calculating Adomian polynomials. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Nonlinear PDE; Huxley equation; Adomian decomposition method; Variational iteration method; Lagrange multiplier
1. Introduction Nonlinear phenomena are of fundamental importance in various fields of science and engineering. Most nonlinear models of real-life problems are still very difficult to solve either numerically or theoretically. Many assumptions have to be made unnecessarily to make nonlinear models solvable. The generalized Huxley equation ut uxx ¼ buð1 ud Þðud cÞ;
0 6 x 6 1;
tP0
ð1Þ
with the initial condition hc c i1=d uðx; 0Þ ¼ þ tanhðrcxÞ ð2Þ 2 2 describes nerve pulse propagation in nerve fibres and wall motion in liquid crystals. The exact solution was derived by Wang et al. [1] using nonlinear transformations and is given by 1=d c c ð1 þ d cÞq þ tanh rc x þ uðx; tÞ ¼ ; ð3Þ t 2 2 2ð1 þ dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r = dq/4(1 + d) and q ¼ 4bð1 þ dÞ.
*
Corresponding author. E-mail address: [email protected] (B. Batiha).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.07.166
B. Batiha et al. / Applied Mathematics and Computation 186 (2007) 1322–1325
1323
In order to solve Eq. (1) numerically, many researchers have used various numerical methods. Authors studied Adomian decomposition method for generalized Huxley equation [2], Wazwaz [3] studied generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Wang [1] studied the solitary wave solutions of the generalized Burger’s-Huxley equation, Hashem et al. [4] studied generalized Burger’s- Huxley equation, and Estevez [5] presented non-classical symmetries and the singular modified the Burger’s and Burger’s-Huxley equation. In this paper, generalized Huxley equation solved by He’s Variational Iteration Method. The numerical results are compared with the exact solutions and that obtained previously by the Adomian decomposition method [2]. 2. Variational iteration method To illustrate the basic concepts of the VIM, we consider the following nonlinear differential equation Lu þ Nu ¼ gðxÞ; ð4Þ where L is a linear operator, N is a nonlinear operator, and g(x) is an inhomogeneous term. According to the VIM [6–11], we can construct a correction functional as follows: Z x unþ1 ðxÞ ¼ un ðxÞ þ kfLun ðsÞ þ N ~ uðsÞ gðsÞgds; ð5Þ 0
where k is a general Lagrangian multiplier [6–8,12] which can be identified optimally via the variational theory, the subscript n denotes the nth-order approximation and ~un is considered as a restricted variation [6–9], i.e. e~ un ¼ 0. In the next section, the variation iteration method has been successfully used to study generalized Huxley equation, and the obtained results are consistent with that found by the Adomian decomposition method. 3. Applications To solve Eq. (1) with an initial condition (2) by means of variational iteration method, we construct a correctional functional which reads Z t unþ1 ðx; tÞ ¼ un ðx; tÞ þ kfunt ~ unxx b~ un ð1 ~udn Þð~udn cÞgds; ð6Þ 0
where e~ un is considered as restricted variations, i.e. e~un ¼ 0. Its stationary conditions are given by k0 ðsÞ ¼ 0;
1 þ kðsÞjs¼t ¼ 0;
ð7Þ
the Lagrange multipliers, therefore, can be identified as k = 1, and the variational iteration formula is given by Z t unþ1 ðx; tÞ ¼ un ðx; tÞ unt unxx bun ð1 udn Þðudn cÞds: ð8Þ 0
Taking the initial approximation u(x, 0) as given by (2), the iteration formula (8) yields uðx; 0Þ ¼ Bm ; u1 ðx; tÞ ¼ Bm þ
1 n þ 12 Am c4 C 2 1 m1 3 1 m2 4 2 2 2 m md md
2 B c ACr t B c C r t þ bB ð1 B ÞðB cÞt: 2 1 4 d nþ A d 4d 2
ð9Þ
where A = tanh(rcx), B = n(1 + A), C = 1 A2, m = 1/d and n = c/2. The rest of the components of the iteration formula (6) can be obtained conveniently using the computer algebra package like Maple. 4. Numerical results and discussion We shall illustrate the accuracy and efficiency of variational iteration method applied to Eq. (1) compared to the Adomian decomposition method [2] applied to same equation. For this purpose, we consider the same
1324
B. Batiha et al. / Applied Mathematics and Computation 186 (2007) 1322–1325
Table 1 Numerical solutions for b = 1, c = 0.001 and d = 1 x
t
Exact
ADM [2]
VIM
0.1
0.05 0.1 1
5.00030171E04 5.00042665E04 5.00267553E04
5.00005184E04 4.99992690E04 4.99767803E04
5.00005184E04 4.99992690E04 4.99767803E04
0.5
0.05 0.1 1
5.00100882E04 5.00113376E04 5.00338263E04
5.00075895E04 5.00063401E04 4.99838513E04
5.00075895E04 5.00063401E04 4.99838513E04
0.9
0.05 0.1 1
5.00171593E04 5.00184087E04 5.00408974E04
5.00146605E04 5.00134111E04 4.99909224E04
5.00146605E04 5.00134111E04 4.99909224E04
Table 2 Numerical solutions for b = 1, c = 0.001 and d = 2 x
t
Exact
ADM [2]
VIM
0.1
0.05 0.1 1
2.23618841E02 2.23624429E02 2.23724988E02
2.23607664E02 2.23602076E02 2.23501462E02
2.23607664E02 2.23602076E02 2.23501490E02
0.5
0.05 0.1 1
2.23644658E02 2.23650245E02 2.23750792E02
2.23633483E02 2.23627895E02 2.23527292E02
2.23633483E02 2.23627895E02 2.23527320E02
0.9
0.05 0.1 1
2.23670472E02 2.23676058E02 2.23776594E02
2.23659298E02 2.23653711E02 2.23553120E02
2.23659298E02 2.23653711E02 2.23553148E02
Table 3 Numerical solutions for b = 1, c = 0.001 and d = 3 x
t
Exact
ADM [2]
VIM
0.1
0.05 0.1 1
7.93740204E02 7.93760039E02 7.94116901E02
7.93700531E02 7.93680693E02 7.93323439E02
7.93700531E02 7.93680695E02 7.93323637E02
0.5
0.05 0.1 1
7.93819558E02 7.93839389E02 7.94196179E02
7.93779893E02 7.93760059E02 7.93402876E02
7.93779894E02 7.93760061E02 7.93403074E02
0.9
0.05 0.1 1
7.93898897E02 7.93918724E02 7.94275442E02
7.93859239E02 7.93839409E02 7.93482298E02
7.93859240E02 7.93839411E02 7.93482496E02
parameter values for the generalized Huxley equation (1) as considered specifically in [2], we take b = 1, c = 0.001 and d = 1, 2, 3. We present in Tables 1–3, the values of exact solution, five-term approximate of ADM and 1-iteration VIM. The results clearly show that VIM is more efficient than the ADM. The VIM avoids the needs for calculating the Adomian polynomials which can be difficult in some cases. 5. Conclusion In this paper, the variational iteration method has been successfully used to study generalized Huxley equation. The solution obtained by means of the variational iteration method is an infinite power series for appro-
B. Batiha et al. / Applied Mathematics and Computation 186 (2007) 1322–1325
1325
priate initial condition, which can be, in turn, expressed in a closed form. The exact solutions are compared with that found by others by using Adomian decomposition method. The results show that the variational iteration method is a powerful mathematical tool for finding the exact and numerical solutions of nonlinear equations. In our work we use the Maple Package to calculate the series obtained from the variational iteration method. Acknowledgement The authors would like to acknowledge the financial support received from the Academy of Sciences Malaysia under the SAGA grant no. P24c. References [1] X.Y. Wang, Z.S. Zhu, Y.K. Lu, Solitary wave solutions of the generalized Burgers–Huxley equation, Phys. Lett. A 23 (1990) 271–274. [2] I. Hashim, M.S.M. Noorani, B. Batiha, A note on the Adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput., in press. doi:10.1016/j.amc.2006.03.011. [3] A.M. Wazwaz, Travelling wave solutions of generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations, Appl. Math. Comput. 169 (2005) 639–656. [4] I. Hashim, M.S.M. Noorani, M.R. Said Al-Hadidi, Solving the generalized Burgers–Huxley equation using the Adomian decomposition method, Math. Comput. Model 43 (2006) 1404–1411. [5] P.G. Estevez, Non-classical symmetries and the singular modified Burger’s and Burger’s–Huxley equation, Phys. Lett. A 27 (1994) 2113–2127. [6] J.H. He, A new approach to Nonlinear Partial Differential Equations, Comm. Nonlinear Sci. Numer. Simul. 2 (4) (1997) 230–235. [7] J.H. He, Variational iteration method – a kind of non-linear analytical technique: some examples, J. Non-linear Mech. 34 (1999) 699– 708. [8] J.H. He, Variational iteration method for delay differential equations, Comm. Nonlinear Sci. Numer. Simul. 2 (4) (1997) 235–236. [9] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burger’s and coupled Burger’s equation, J. Comput. Appl. Math. 181 (2) (2005) 245–251. [10] S. Momani, S. Abuasad, Application of He’s variational iteration method to Helmholtz equation, Chaos, Soliton and Fractals 27 (5) (2006) 1119–1123. [11] A.A. Soliman, A numerical simulation and explicit solutions of KdV–Burgers and Lax’s seventh-order KdV equation, Chaos, Soliton and Fractals 29 (2) (2006) 294–302. [12] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat-Nassed (Ed.), Variational Method in the Mechanics of Solids, Pergamon Press, 1978, pp. 156–162.
Numerical simulation of the generalized Huxley equation by He’s variational iteration method B. Batiha *, M.S.M. Noorani, I. Hashim School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, Malaysia
Abstract By means of variational iteration method the solution of generalized Huxley equation are obtained, comparison with the Adomian decomposition method is made, showing that the former is more effective than the later. In this paper, He’s variational iteration method is introduced to overcome the difficulty arising in calculating Adomian polynomials. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Nonlinear PDE; Huxley equation; Adomian decomposition method; Variational iteration method; Lagrange multiplier
1. Introduction Nonlinear phenomena are of fundamental importance in various fields of science and engineering. Most nonlinear models of real-life problems are still very difficult to solve either numerically or theoretically. Many assumptions have to be made unnecessarily to make nonlinear models solvable. The generalized Huxley equation ut uxx ¼ buð1 ud Þðud cÞ;
0 6 x 6 1;
tP0
ð1Þ
with the initial condition hc c i1=d uðx; 0Þ ¼ þ tanhðrcxÞ ð2Þ 2 2 describes nerve pulse propagation in nerve fibres and wall motion in liquid crystals. The exact solution was derived by Wang et al. [1] using nonlinear transformations and is given by 1=d c c ð1 þ d cÞq þ tanh rc x þ uðx; tÞ ¼ ; ð3Þ t 2 2 2ð1 þ dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r = dq/4(1 + d) and q ¼ 4bð1 þ dÞ.
*
Corresponding author. E-mail address: [email protected] (B. Batiha).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.07.166
B. Batiha et al. / Applied Mathematics and Computation 186 (2007) 1322–1325
1323
In order to solve Eq. (1) numerically, many researchers have used various numerical methods. Authors studied Adomian decomposition method for generalized Huxley equation [2], Wazwaz [3] studied generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Wang [1] studied the solitary wave solutions of the generalized Burger’s-Huxley equation, Hashem et al. [4] studied generalized Burger’s- Huxley equation, and Estevez [5] presented non-classical symmetries and the singular modified the Burger’s and Burger’s-Huxley equation. In this paper, generalized Huxley equation solved by He’s Variational Iteration Method. The numerical results are compared with the exact solutions and that obtained previously by the Adomian decomposition method [2]. 2. Variational iteration method To illustrate the basic concepts of the VIM, we consider the following nonlinear differential equation Lu þ Nu ¼ gðxÞ; ð4Þ where L is a linear operator, N is a nonlinear operator, and g(x) is an inhomogeneous term. According to the VIM [6–11], we can construct a correction functional as follows: Z x unþ1 ðxÞ ¼ un ðxÞ þ kfLun ðsÞ þ N ~ uðsÞ gðsÞgds; ð5Þ 0
where k is a general Lagrangian multiplier [6–8,12] which can be identified optimally via the variational theory, the subscript n denotes the nth-order approximation and ~un is considered as a restricted variation [6–9], i.e. e~ un ¼ 0. In the next section, the variation iteration method has been successfully used to study generalized Huxley equation, and the obtained results are consistent with that found by the Adomian decomposition method. 3. Applications To solve Eq. (1) with an initial condition (2) by means of variational iteration method, we construct a correctional functional which reads Z t unþ1 ðx; tÞ ¼ un ðx; tÞ þ kfunt ~ unxx b~ un ð1 ~udn Þð~udn cÞgds; ð6Þ 0
where e~ un is considered as restricted variations, i.e. e~un ¼ 0. Its stationary conditions are given by k0 ðsÞ ¼ 0;
1 þ kðsÞjs¼t ¼ 0;
ð7Þ
the Lagrange multipliers, therefore, can be identified as k = 1, and the variational iteration formula is given by Z t unþ1 ðx; tÞ ¼ un ðx; tÞ unt unxx bun ð1 udn Þðudn cÞds: ð8Þ 0
Taking the initial approximation u(x, 0) as given by (2), the iteration formula (8) yields uðx; 0Þ ¼ Bm ; u1 ðx; tÞ ¼ Bm þ
1 n þ 12 Am c4 C 2 1 m1 3 1 m2 4 2 2 2 m md md
2 B c ACr t B c C r t þ bB ð1 B ÞðB cÞt: 2 1 4 d nþ A d 4d 2
ð9Þ
where A = tanh(rcx), B = n(1 + A), C = 1 A2, m = 1/d and n = c/2. The rest of the components of the iteration formula (6) can be obtained conveniently using the computer algebra package like Maple. 4. Numerical results and discussion We shall illustrate the accuracy and efficiency of variational iteration method applied to Eq. (1) compared to the Adomian decomposition method [2] applied to same equation. For this purpose, we consider the same
1324
B. Batiha et al. / Applied Mathematics and Computation 186 (2007) 1322–1325
Table 1 Numerical solutions for b = 1, c = 0.001 and d = 1 x
t
Exact
ADM [2]
VIM
0.1
0.05 0.1 1
5.00030171E04 5.00042665E04 5.00267553E04
5.00005184E04 4.99992690E04 4.99767803E04
5.00005184E04 4.99992690E04 4.99767803E04
0.5
0.05 0.1 1
5.00100882E04 5.00113376E04 5.00338263E04
5.00075895E04 5.00063401E04 4.99838513E04
5.00075895E04 5.00063401E04 4.99838513E04
0.9
0.05 0.1 1
5.00171593E04 5.00184087E04 5.00408974E04
5.00146605E04 5.00134111E04 4.99909224E04
5.00146605E04 5.00134111E04 4.99909224E04
Table 2 Numerical solutions for b = 1, c = 0.001 and d = 2 x
t
Exact
ADM [2]
VIM
0.1
0.05 0.1 1
2.23618841E02 2.23624429E02 2.23724988E02
2.23607664E02 2.23602076E02 2.23501462E02
2.23607664E02 2.23602076E02 2.23501490E02
0.5
0.05 0.1 1
2.23644658E02 2.23650245E02 2.23750792E02
2.23633483E02 2.23627895E02 2.23527292E02
2.23633483E02 2.23627895E02 2.23527320E02
0.9
0.05 0.1 1
2.23670472E02 2.23676058E02 2.23776594E02
2.23659298E02 2.23653711E02 2.23553120E02
2.23659298E02 2.23653711E02 2.23553148E02
Table 3 Numerical solutions for b = 1, c = 0.001 and d = 3 x
t
Exact
ADM [2]
VIM
0.1
0.05 0.1 1
7.93740204E02 7.93760039E02 7.94116901E02
7.93700531E02 7.93680693E02 7.93323439E02
7.93700531E02 7.93680695E02 7.93323637E02
0.5
0.05 0.1 1
7.93819558E02 7.93839389E02 7.94196179E02
7.93779893E02 7.93760059E02 7.93402876E02
7.93779894E02 7.93760061E02 7.93403074E02
0.9
0.05 0.1 1
7.93898897E02 7.93918724E02 7.94275442E02
7.93859239E02 7.93839409E02 7.93482298E02
7.93859240E02 7.93839411E02 7.93482496E02
parameter values for the generalized Huxley equation (1) as considered specifically in [2], we take b = 1, c = 0.001 and d = 1, 2, 3. We present in Tables 1–3, the values of exact solution, five-term approximate of ADM and 1-iteration VIM. The results clearly show that VIM is more efficient than the ADM. The VIM avoids the needs for calculating the Adomian polynomials which can be difficult in some cases. 5. Conclusion In this paper, the variational iteration method has been successfully used to study generalized Huxley equation. The solution obtained by means of the variational iteration method is an infinite power series for appro-
B. Batiha et al. / Applied Mathematics and Computation 186 (2007) 1322–1325
1325
priate initial condition, which can be, in turn, expressed in a closed form. The exact solutions are compared with that found by others by using Adomian decomposition method. The results show that the variational iteration method is a powerful mathematical tool for finding the exact and numerical solutions of nonlinear equations. In our work we use the Maple Package to calculate the series obtained from the variational iteration method. Acknowledgement The authors would like to acknowledge the financial support received from the Academy of Sciences Malaysia under the SAGA grant no. P24c. References [1] X.Y. Wang, Z.S. Zhu, Y.K. Lu, Solitary wave solutions of the generalized Burgers–Huxley equation, Phys. Lett. A 23 (1990) 271–274. [2] I. Hashim, M.S.M. Noorani, B. Batiha, A note on the Adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput., in press. doi:10.1016/j.amc.2006.03.011. [3] A.M. Wazwaz, Travelling wave solutions of generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations, Appl. Math. Comput. 169 (2005) 639–656. [4] I. Hashim, M.S.M. Noorani, M.R. Said Al-Hadidi, Solving the generalized Burgers–Huxley equation using the Adomian decomposition method, Math. Comput. Model 43 (2006) 1404–1411. [5] P.G. Estevez, Non-classical symmetries and the singular modified Burger’s and Burger’s–Huxley equation, Phys. Lett. A 27 (1994) 2113–2127. [6] J.H. He, A new approach to Nonlinear Partial Differential Equations, Comm. Nonlinear Sci. Numer. Simul. 2 (4) (1997) 230–235. [7] J.H. He, Variational iteration method – a kind of non-linear analytical technique: some examples, J. Non-linear Mech. 34 (1999) 699– 708. [8] J.H. He, Variational iteration method for delay differential equations, Comm. Nonlinear Sci. Numer. Simul. 2 (4) (1997) 235–236. [9] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burger’s and coupled Burger’s equation, J. Comput. Appl. Math. 181 (2) (2005) 245–251. [10] S. Momani, S. Abuasad, Application of He’s variational iteration method to Helmholtz equation, Chaos, Soliton and Fractals 27 (5) (2006) 1119–1123. [11] A.A. Soliman, A numerical simulation and explicit solutions of KdV–Burgers and Lax’s seventh-order KdV equation, Chaos, Soliton and Fractals 29 (2) (2006) 294–302. [12] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat-Nassed (Ed.), Variational Method in the Mechanics of Solids, Pergamon Press, 1978, pp. 156–162.